Performance-Guaranteed Solutions for Multi-Agent Optimal Coverage Problems using Submodularity, Curvature, and Greedy Algorithms

We consider a class of multi-agent optimal coverage problems where the goal is to determine the optimal placement of a group of agents in a given mission space such that they maximize a joint ``coverage'' objective. This class of problems is extremely challenging due to the non-convex nature of the mission space and of the coverage objective. With this motivation, we propose to use a greedy algorithm as a means of getting feasible coverage solutions efficiently. Even though such greedy solutions are suboptimal, the submodularity (diminishing returns) property of the coverage objective can be exploited to provide performance bound guarantees - not only for the greedy solutions but also for any subsequently improved solutions. Moreover, we show that improved performance bound guarantees (beyond the standard (1-1/e) performance bound) can be established using various curvature measures that further characterize the considered coverage problem. In particular, we provide a brief review of all existing popular curvature measures found in the submodular maximization literature, including a recent curvature measure that we proposed, and discuss in detail their applicability, practicality, and effectiveness in the context of optimal coverage problems. Moreover, we characterize the dependence of the effectiveness of different curvature measures (in providing improved performance bound guarantees) on the agent sensing capabilities. Finally, we provide several numerical results to support our findings and propose several potential future research directions.